Durham Research Online - COnnecting REpositoriesa fundamentally and philosophically different “top-down” ap-proach, to the traditional inversion, and it relies entirely on forward

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Roberts, A.W. and Hobbs, R.W. and Goldstein, M. and Moorkamp, M. and Jegen, M. and Heincke, B. (2016)'Joint stochastic constraint of a large data set from a salt dome.', Geophysics., 81 (2). ID1-ID24.

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Joint stochastic constraint of a large data set from a salt dome

Alan W. Roberts1, Richard W. Hobbs2, Michael Goldstein3, Max Moorkamp4,Marion Jegen5, and Bjørn Heincke6

ABSTRACT

Understanding the uncertainty associated with large joint geo-physical surveys, such as 3D seismic, gravity, and magnetotellu-ric (MT) studies, is a challenge, conceptually and practically. Bydemonstrating the use of emulators, we have adopted a MonteCarlo forward screening scheme to globally test a prior modelspace for plausibility. This methodology means that the incorpo-ration of all types of uncertainty is made conceptually straight-forward, by designing an appropriate prior model space, uponwhich the results are dependent, from which to draw candidatemodels. We have tested the approach on a salt dome target, over

which three data sets had been obtained; wide-angle seismic re-fraction, MT and gravity data. We have considered the data setstogether using an empirically measured uncertain physical rela-tionship connecting the three different model parameters: seismicvelocity, density, and resistivity, and we have indicated the valueof a joint approach, rather than considering individual parametermodels. The results were probability density functions over themodel parameters, together with a halite probability map. Theemulators give a considerable speed advantage over runningthe full simulator codes, and we consider their use to have greatpotential in the development of geophysical statistical constraintmethods.

INTRODUCTION

To map a region of earth, it is commonplace to use one or morekinds of data sets to constrain structural models parameterized byone or more proxy parameters, such as seismic velocity, density, orresistivity. An interpreter will then use their geologic insight com-bined with these models to make judgments about the region. Thismay be with a view to, for example, determining where appropriatedrilling locations might lie to maximize the possibility of hydrocar-bon extraction. There are many approaches used to constrain theproxy models, ranging from deterministic inverse approaches toMarkov chain Monte Carlo (MCMC) search schemes (Press, 1970;Sambridge and Mosegaard, 2002; Shapiro and Ritzwoller, 2002;Hobro et al., 2003; Gallardo and Meju, 2004; Roy et al., 2005;Heincke et al., 2006; Meier et al., 2007; Moorkamp et al., 2011).Deterministic inverse schemes are optimal when the uncertainties in

the data and physical system are small, and the aim is to find theoptimum model as fast as possible. This approach works by re-peated model update so as to minimize the difference between theobserved data and the simulator’s output. However, in many scenar-ios, there are considerable uncertainties associated with the data andphysics concerned. In this case, statistical schemes may be adopted.In these methods, the aim is normally to discern the entire plausiblemodel space for the system concerned. The character of such stat-istical schemes varies from the entirely forward-based screeningmethod (Press, 1970), to the more targeted sampling strategy of theMCMC approach (Hastings, 1970; Sambridge and Mosegaard,2002). MCMC schemes seek to sample enough of the model spaceto give a robust uncertainty estimate; however, often, the numberof forward simulations in both of these methods required to suffi-ciently sample the space for large systems often makes these meth-ods computationally impracticable. Thus, often in part due to the

Manuscript received by the Editor 27 February 2015; revised manuscript received 21 August 2015; published online 19 February 2016.1Formerly Durham University, Department of Earth Sciences, Durham, UK, and Durham University, Department of Mathematics, Durham, UK; presently

Geospatial Research Limited, Durham, UK. E-mail: [email protected] University, Department of Earth Sciences, Durham, UK. E-mail: [email protected] University, Department of Mathematics, Durham, UK. E-mail: [email protected] GEOMAR, Kiel, Germany; presently University of Leicester, Department of Geology, Leicester, UK. E-mail: [email protected], Kiel, Germany. E-mail: [email protected] GEOMAR, Kiel, Germany; presently GEUS, Department of Petrology and Economic Geology, Copenhagen, Denmark. E-mail: [email protected].© 2016 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 81, NO. 2 (MARCH-APRIL 2016); P. ID1–ID24, 21 FIGS., 4 TABLES.10.1190/GEO2015-0127.1

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lack of a feasible method of assessing the uncertainty associatedwith a system and also conceptual difficulties in incorporating aparticular kind of uncertainty into the constraint process, the uncer-tainty assessments that are fed into the decision-making processescan be ill informed.The approach adopted here is a forward-modeling-based screen-

ing strategy. To make the process computationally feasible, we build“emulators” for each of the forward modeling (“simulator”) codes,trained over the prior model space. We use the term emulator tomean a statistical model of the output generated by a complex for-ward modeling code, or simulator. The aim of building an emulatoris to have a means of generating a fast uncertainty calibrated esti-mate of the simulator output (which maybe be time expensive tocompute). Through providing uncertainty-calibrated rapid estimatesof the full simulator output, these emulators overcome the computa-tional barrier of making vast numbers of complex simulator runs.By iteratively rejecting an implausible model space and updatingthe emulators, the plausible model space is discerned. In this study,we apply the method to constrain a region of earth characterized bya salt dome using three kinds of data: seismic refraction, magneto-telluric (MT), and gravity data sets for a 1D seven-layered parameter-ization, and we construct a rock-type probability map based on thefractional salt versus sediment model acceptance for the region. Asdiscussed in Osypov et al. (2011) and elsewhere, proper assessmentof risk in hydrocarbon exploration requires not only the analysis of aproxy-parameter model, but also a full analysis of the structural un-certainty. The ability to construct a probability map in this mannerhas the potential to be of considerable value in this regard.

Joint inversion

Deterministic inversion methods (Tarantola, 2005), in which theaim is to iteratively update a model so as to reduce some objectivefunction, are commonly used when the data come from a singletechnique. However, using such schemes in a joint framework inwhich the relationship between the physical parameters (e.g., seis-mic velocity, resistivity, and density) is empirical and uncertain,poses philosophical challenges regarding the coupling strategy,for example, the weighting attached to maintain structural coher-ency across the various methods (Gallardo and Meju, 2004). Sim-ilarly, there are also conceptual intricacies associated with properlyand quantitatively including most kinds of uncertainty associatedwith the problem, for example, uncertainty in the data measure-ments and model discrepancy (due to the fact that a model is nota complete representation of nature). Recently however, a few au-thors such as Roy et al. (2005), Heincke et al. (2006), and Moor-kamp et al. (2011, 2013) make considerable progress in developingstructural coupling-based joint inversion methodologies throughcrossgradient and other coupling schemes. Bodin et al. (2012) alsodevelop hierarchical Bayes approaches for joint inversion.

Statistical schemes

Statistical schemes designed to assess uncertainty, such as simu-lated annealing, genetic algorithms, and MCMC approaches, can beused when the number of model parameters is small. Sambridge andMosegaard (2002) give a useful review of the varied methods thatcan be used and their historical development.However, as is commented in Sambridge and Mosegaard (2002),

if the number of parameters is large, then these methods become

unfeasible because the number of complex, and possibly expensive,forward model simulations becomes impracticably large given thecomputation time required. In a few scientific fields, such as clima-tology, volcanic hazard prediction, ocean modeling, and cosmology(Logemann et al., 2004; Rougier, 2008; Bayarri et al., 2009; Vernonand Goldstein, 2009), in which forward simulators are also highlytime expensive to run, emulators are often used. An emulator is astatistical representation of the forward modeling simulator, whichgives a very rapid prediction of the simulator output, with a cali-brated uncertainty.Building an emulator is similar to building a neural network.

Neural networks are successfully used to solve inverse problemsin geophysics, for example, Meier et al. (2007), who develop a neu-ral network system to invert S-wave data. Others have also devel-oped methods of using quick approximations to a full forward codein inversion schemes, for example, James and Ritzwoller (1999),who use truncated perturbation expansions to approximate Ray-leigh-wave eigenfrequencies and eigenfunctions, and Shapiro andRitzwoller (2002), who take a similar methodology in an MCMCscheme to construct a global shear-velocity model of the crust andupper mantle. In each of these cases, the aim is to minimize someobjective function or maximize a likelihood function.Here, we adopt a statistical approach that is fundamentally differ-

ent in that it is based entirely on forward modeling, as opposed tousing any kind of objective/likelihood function or inverse step. Wesimply seek to discern which areas of model space are plausible andwhich are implausible, given the observed data. This approach hasbeen proposed in the past (e.g., Press, 1970), and it is used in a varietyof settings such as the history matching of hydrocarbon reservoir pro-duction data (Murtha, 1994; Li et al., 2012). However, in the contextof structural constraint, it is largely sidelined in favor of more search-efficient schemes such as those described above. We implement thisforward approach by the use of emulators to make it more computa-tionally efficient. Roberts et al. (2012) describe our methodology fora synthetic scenario; however, here we describe a number of mod-ifications to achieve greater stability and efficiency. We developand apply the approach to observed 3D joint seismic, MT, and gravitydata sets obtained from a salt dome region, and we ultimately deter-mine a model probability map for the profile. The method is akin tothe response surface methodologies beginning to be used in the fieldof reservoir simulation (Zubarev, 2009). However in this case, weseek to fully model the uncertainty in the simulator-prediction sys-tem, and hence we aim to construct response clouds, rather than sur-faces. The method is shown diagrammatically in Figure 1.The strategy here, to exclude model space, rather than build up

the plausible space searching from some starting model, representsa fundamentally and philosophically different “top-down” ap-proach, to the traditional inversion, and it relies entirely on forwardcomputation. Because we globally sample the prior plausible modelspace, seeking to exclude implausible model space, rather thansearching a part of the model space for plausible models, the un-certainty measures which are obtained are maxima, rather than min-ima, given the prior model space and choices of tuning parametersmade in the analysis.Building statistical system models, or emulators, successively in

a multicycle fashion, we progressively refine the plausible space.Because a proper consideration of uncertainty in an inverse schemecan be conceptually difficult, often, when any consideration ofuncertainty is made, it is commonly specified to be Gaussian in

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character at times not because it is indeed Gaussian, but simply be-cause of mathematical convenience. The fact that, in our method,the screening process relies entirely on forward modeling meansthat it is conceptually straightforward to include uncertainty pertain-ing to any part of the system by building the appropriate distributionover the prior model space. These uncertainties may take the form ofdata uncertainty, physical uncertainty, model discrepancy, or others,which in an inversion scheme, including physical uncertainties andmodel discrepancy, among others, can be conceptually difficult.

Emulation

To overcome computational limitations in our forward screeningMonte Carlo scheme, we build and use emulators (Kennedy andO’Hagan, 2001; Vernon et al., 2009). Like the case of a neural net-work, an emulator is designed using training models and data sets,and it seeks to predict the output data arising for a given modelparameter set. However, an emulator differs from a neural networkin that it seeks to not only predict the output of a system from aninput, but also to do so with a fully calibrated uncertainty. An em-ulator treats the parametric and nonparametric parts of the systemholistically, giving a full stochastic representation of the system.Because of this focus on uncertainty calibration, emulators can beused to rapidly screen model space for implausibility (Goldstein andWooff, 2007). This would not be the case with an uncalibrated sim-ulator-prediction system because there is no measure or criterion todiscern whether a comparative data set is sufficiently close to the

observed data set to be deemed plausible. Although the method de-scribed here is very much a forward modeling philosophy, in thatwe are simply seeking to trial sets of model parameters for plausibil-ity, one could consider that the fitting of parametric functions tobuild the emulator model of the forward simulator constitutes apartly inverse component. However, because the forward simulatoritself, rather than the model parameters is being “inverted” for, theemulator screening method is fundamentally different to a tradi-tional inversion scheme.Although there are occasional instances of emulators being de-

veloped for earth systems (Logemann et al., 2004), they have notbeen widely applied in the geosciences. Here, we review and dem-onstrate the use of an emulator (Roberts et al., 2010, 2012) to con-strain the structure of a salt diapir using 1D profiles through a 3Djoint data set. Figure 1 summarizes the strategy adopted in this study.The data consist of 3D seismic data, full tensor gravity (FTG) data,and MT data from a salt dome. Examples from the three data sets areshown in Figure 2.

THEORY AND PRELIMINARIES

In performing an experiment to test a model, the scientist has a setof output data points, a set of model parameters, and a function(simulator or forward modeling code) f, which defines the relation-ship between the model parameters θ and the “perfect” data ψ(equation 1) as follows:

Fit datasets to model parametersusing suitable simple + fast functions

*Calibrate uncertainty by comparingsimulator outputs to prediction based

on reconstruction using trainingmodels and fitted coefficients

Set of coefficients which can be usedto predict a simulator output for a

given set of model parameters

Uncertainty function associatedwith emulator prediction

EMULATORSGenerate candidate

model

EMULATORS

Approximate dataprediction and emulator

uncertainty

Does approximatepredicted datasets lie withineach emulator uncertainty

Store candidate model asplausible

YES

Discard candidate model

Have N plausible modelsbeen generated?

NO

YES

Compare uncertainty functionto uncertainty function from

previous cycle.

First cycle?

YES

NO

STOPAll discernible

structure has nowbeen extracted from

the system.

START

YES

NO

Seismic, MT, gravitysimulator codes

Define prior model space over all activeparameters and processes using

Generate *1500 training MODELS for eachof the seismic, MT and gravity domains

*1500 training DATASETS for each ofthe seismic, MT and gravity domains

*The value of 1500 is determined on the first cycleby observing when the uncertainty function is no

longer updated when more models are added to thecalibration step. At this point, the model space has

been sufficiently sampled by training models

NO

THE OUTPUTThe distribution of all plausible models given our

beliefs about the physics of the system, ouruncertainties about those beliefs, the dataset, datameasurement uncertainty and model discrepancy.

PRELIMINARIES

TRAININGSIMULATIONS

BUILDINGEMULATORS

SCREEN MODEL SPACE

THE INPUTS

Seismic, MT and gravitysimulator codes

Seismic, MT andgravity datasets

Observed seismic,MT and gravity data

Geophysical insight

Local geophysical insight (for priormodel space and empirical inter-

parameter relationships)

Tuning parameters suchas implausibility criteriaand number of modelsrequired for each cycle

Figure 1. The emulator screening methodology.

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ψ ¼ fðθÞ: (1)

In the case of a heavily parameterized system, with many nodesin the parameter models, and many output data, this function, orsimulator f can take a long time to evaluate. For a typical inversionproblem involving seismic data, many thousands of evaluations of f

maybe required, and the problem very quickly becomes impracticalto solve if the time spent evaluating f is significant.Vernon et al. (2009) and Kennedy and O’Hagan (2001) use em-

ulators to address this kind of problem in which large numbers ofcomplex simulator evaluations are required. An emulator seeks torepresent the simulator function f as a combination of a computa-tionally cheap deterministic function (e.g., a polynomial) h and aGaussian process g (Rasmussen and Williams, 2010):

ψ ¼ hðθÞ þ gðθÞ: (2)

The aim is not to completely replace the full simulator, but todevelop a system such that one can very quickly glean enough in-formation from the relationship between the model parameters andoutput data to make meaningful judgments about whether regionsof model space can be excluded from the analysis on the basis thatthey would result in output data not compatible with the ob-served data.Because h and g are fast to evaluate, a considerable time savings

(of orders of magnitude) can typically be achieved by this approach.In the study detailed here, we adopt a multistage approach (Vernonet al., 2009) of seeking to describe the global behavior and then, asthe implausible model space is excluded, to describe increasinglylocalized behavior as we develop more predictively accurate emu-lators.

DATA, MODEL SPACE, AND THE INVERSEPROBLEM

Data

A joint data set for this study was kindly supplied by Statoil. It isa joint 3D seismic, FTG and MT data set recorded over a regionknown to Statoil as being characterized by a salt diapiric body (Fig-ure 2). To simplify the problem, we enforce a local 1D solution. TheMT data were transformed into the directionally independent Ber-dichevsky invariant form (Berdichevsky and Dmitriev, 2002), seis-mic data were picked for 1868 shot gathers and transformed into thecommon midpoint (CMP) domain, and the closest CMP profiles toeach MT station were identified and used as 1D seismic data for thepurposes of the study. The FTG data were transformed to scalardata, and the closest measurement to each MT station was identi-fied. Results are thus generated for the series of 1D seismic, MT,and gravity data sets collocated at the site of each MT station. Inthis paper, each site is labeled “STxx”, where “xx” can take thevalue 1–14, for example, in Figure 2.

Gravity datum

In addition, it was also necessary to establish a datum for thegravity data measurements so as to make meaningful comparisonbetween each station. This is because if the models are allowed tobe of arbitrary total thickness, then the gravity reading could beconsidered as simply as a free parameter and afford no constraint.In practice, this is an expression of the Airy hypothesis of isostasy(Airy, 1855). This calibration requires the tying of the measuredgravity point at one station to the simulator output at that station,with an assumption about the structure at that station, against whichresults at the other stations can be considered as being relativeto. This assumption might be that the model is of a given total

Offset

Tim

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1 km

1 s

Scale

a)

log(MT7r)

log(MT7i)

log(MT12r)

log(MT12i)

log (frequency)10

log

(Z)

10

–4 –3 –2 –1 0

0

–1

1c)

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Rel

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0.2 0.4 0.6 0.8Relative longitude

ST7 ST12

–70

–60

–50

–40

–30

–20

–10

0

10

20

30

40

Fre

e ai

r gr

avity

ST5

ST13

b)

2D profile

Figure 2. Data examples: (a) seismic, (b) gravity, and (c) MT. Thered dots on the seismic gather show the first arrival wide-angle turn-ing waves, which are being modeled in this study. On the gravitymap in panel (b), the locations of ST5, ST7, ST12, and ST13 (whichare frequently referred to in this study) are marked with purple stars,and the track of the 2D line for which profiles are shown in laterfigures. The MT data plot shows Berdichevsky invariant (Berdi-chevsky and Dmitriev, 2002) Re(Z) and Im(z) for stations ST7(red) and ST12 (green).

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thickness, or some other consideration about the model. We chooseto use the assumption that at a given point along the line, there is nosalt present. On the grounds that prior studies indicate that it liesover a region of sediment, we chose to use station ST12 (Figure 2)for the purpose of gravity calibration. The calibration was carriedout by running the code with gravity screening disabled, only gen-erating “sediment models” and then generating density modelsusing the relationship in equation 7 from the plausible velocity mod-els. Gravity measurements were then generated from these modelsusing the gravity simulator, and the most likely gravity measure-ment were compared with the measured gravity value at stationST12. The difference between these two values was then used asa “correction” value for comparing screened gravity values to mea-sured gravity values at all the other stations. In practice, this meansthat the gravity results in this study and the “salt content” are beingmeasured relative to station ST12 screened with the assumption thatthere is no salt there (the prior probability of salt in each layer isset to zero, and thus no salt models are generated). However, for ameaningful comparison (and a test of the assumption that there is nosalt at ST12), the screening process is then repeated for ST12 withthe possibility of salt models included with a probability of 0.5 ineach layer, as was the case for the other stations. A comparisonof the results prior to the gravity calibration (without gravity con-straint) and postcalibration (the main results presented in this paper)would provide an interesting study; however, for brevity, these pre-liminary outputs are not discussed here.

Methods and model space

Our goal is to describe the model in terms of proxy quantities(P-wave velocity, resistivity, and density) and also to obtain a rockprobability map for the profile along which the MT stations are lo-cated. Although the priors in this study are somewhat illustrative, ina scenario in which the priors are well constrained and tested, such aprobability map may be used as a more direct input to evaluate geo-logic risk, rather than simply providing proxy-parameter modelsthat the scientist must then interpret. We discern the distribution ofjointly plausible models with respect to each of the seismic, gravity,andMT data sets, given all of the uncertainties wewish to specify, bygenerating candidate joint models drawn from a prior model space. Inthis way, we effectively screen the model space using the interpara-meter relationship to discriminate between salt and sediment rocks.As is noted in Roberts et al. (2012), the simulators, particularly

the seismic simulator (Heincke et al., 2006), are more sophisticatedthan required for the problem at hand; however, to facilitate futuredevelopment and allow integration and direct comparison with otherwork (Heincke et al., 2006; Moorkamp et al., 2011, 2013), we usethese simulators.

Prior model space

The first consideration is the initial model space within which weconsider the plausible models for the system to lie (Figure 1). Ourfocus here is on the emulation methodology as a means to screenand constrain model space, rather than on generating robust Baye-sian posterior distributions for the particular region used for the casestudy. As such, here we have placed only a cursory emphasis on thedetermination and specification of the prior model space. The finalresult should, therefore, not be considered as a true Bayesian con-straint from which a genuine geologic inference can be made about

this region. For such a result, proper consideration of appropriatepriors should be made, and proper sensitivity calibration throughsampling those priors. Accordingly, the analysis presented hereis made on the assumption that there is indeed halite present in theregion of earth under consideration and on the basis that that halite,and indeed the surrounding material, has properties reflective ofthose seen globally and in conjunction with the borehole data setdescribed below.Similarly, at several points in the emulator building and screening

process, the tuning parameters are set. Again, here these are chosensomewhat qualitatively and arbitrarily. In reality, the choices madefor these parameters also constitute part of the model space, and sofor the results of the screening process to be geologically meaning-ful, expert judgment should be used in the choice of these param-eters, with a prior distribution that can be fully sampled. The finalresults of the analysis presented here should thus be treated as beingillustrative of the method and should be subject to all of the explicitand implict assumptions made, rather than being authoritative as tothe earth structure in question.Our prior joint model space is constrained primarily by three

influences: (1) the interparameter relationship linking the seismicvelocity, density, and resistivity parameters, (2) the range of geo-physically plausible values which each of these parameters mayadopt, and (3) the prior probability of salt existing in each layer.

Physical parameter relationship.—For the purpose describedhere, a rock is characterized by its combination of physical properties(inour case, resistivity, seismicvelocity, anddensity),whichare encap-sulated by the empirical physical parameter relationships that connectthem. In this joint setting,we therefore propose, for a given layer, com-binations ofmodel parameters across each of the domains that are con-nectedbyeitherasediment relationshipora“salt relationship.Bydoingthis, and then assessing the fraction of models deemed plausible gen-erated using each relationship for a given depth, we can then make astatement about the rejection ratio formodels generated using each re-lationship regarding theprobability that salt or sediment exists atdiffer-ent locations and constructing a salt likelihood map for the profile.The interparameter relationship for sediments, although it is em-

pirical and uncertain, may be relatively easily formulated by fittinga curve through well-log data from the survey area (Figure 3). How-ever, the presence of salt complicates the situation somewhat, in thatfor sediment there is a monotonic increase between seismic veloc-ity, density, and resistivity. However, salt has a very characteristicseismic velocity of 4500 m∕s, a density of around 2100 kg∕m3

(Birch, 1966), and very high resistivity (>500 Ωm; Jegen-Kulcsaret al., 2009). Therefore, we define two relationships for our situa-tion, as shown in equations 5–9. In this case, we have chosen theuncertainty to be a function added to a central value. It would alsobe straightforward to specify the uncertainty in other ways, for ex-ample, as uncertainty in the values of the relationship coefficients.In these relationships, r, ρ, and v refer to the resistivity, density, andseismic velocity values, respectively. The value Nða; bÞ refers to asample from a normal distribution of mean a and standard deviationb. The borehole data from which the sediment density/resistivity/velocity relationship was obtained (kindly provided by Statoil) isshown in Figure 3. The borehole is located adjacent to stationST5 (Figure 2). Consideration of Figure 3 suggests that given thatthere are a considerable number of points lying outside the boundsshown for the salt and sediment relationships, there may be a case

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for including a third relationship category of “other,” to accommo-date these currently extremal points more effectively. However, forsimplicity, here we decided to continue with the two-relationshipscheme. In this case, we have used data from this single boreholefor the entire line. A more rigorous study would seek if possible toconsider borehole data from various points along the line to accountfor regional variation.Sediment parameter relationships are given in the following

equations:

log10ðrÞ ¼ −8.72487þ 0.0127274v − 6.4247 × 10−6v2

þ 1.45839 × 10−9v3; (3)

−1.47131×10−13v4þ5.32708×10−18v5þNð0;σrðvÞÞ; (4)

σrðvÞ ¼ −2.931 × 10−2 þ 1.989 × 10−5vþ 1.058 × 10−9v2;

(5)

ρ ¼ −785.68þ 2.09851v − 4.51887 × 10−4v2

þ 3.356 × 10−8v3 þ Nð0; σdðvÞÞ; (6)

and

σρðvÞ ¼ 1.42693 × 102 − 1.11564 × 10−1v

þ 3.0898 × 10−5v2 − 2.52979 × 10−9v3: (7)

Salt parameter relationships are given in the following equations:

log10ðrÞ ¼ 2.8þ Nð0; 0.5Þ (8)

and

ρ ¼ 2073þ Nð0;45Þ: (9)

Parameter ranges.—Another important bound on the model spaceis our belief about the prior plausible model parameter ranges. It isimplicit that the model parameterization should be chosen so as tobe capable of describing the full range of prior plausible modelsusing as few parameters as possible and also that it should be asunique as possible. Here, we consider 1D joint common structuremodels (Jegen-Kulcsar et al., 2009). Thus, we choose a parameter-ization of velocities, densities, and resistivities for a series of layersof common variable thickness. A fuller treatment would involvequantitatively trialing the ability to represent appropriate geologicformations and data sets using a range of numbers of model layersthat is in itself part of the model space definition. In this case, weconsidered that after informal qualitative testing, models parameter-ized by seven layers seemed sufficient in providing the ability forthe system to discern structure, particularly in the shallow regionand the salt body, while not overparameterizing the system giventhe resolution of the observed data sets. The prior model parameterranges for each of these layers are shown in Table 1.

Prior salt probability.—A more subtle constraint on the priormodel space is the prior probability of salt existing in each layer.For this study, we specify this to be 0.5 for each layer. In eachscreening cycle, for each layer, salt or sediment models are gener-ated in the ratio appropriate to the fraction of models (the likelihoodof salt present) deemed plausible for that layer from the pre-vious cycle.

Building an emulator

Having specified the prior model space from which we intend todraw candidate models, we now construct an emulator for each ofthe seismic, gravity, and MT cases. We describe the process in detailfor the seismic case and adopt a similar approach for the MT andgravity cases. The framework for each of these, including the full setof governing equations, is given in Appendix A. The model spaceused for training each emulator was simply defined by the range ofparameter values considered plausible in each of the velocity, resis-tivity, density, and thickness cases (Table 1). In other words, each

0.1

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SaltSediment2

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Figure 3. (a) Resistivity versus velocity and (b) density versus veloc-ity relationships derived from well-log data. The borehole is locatedadjacent to station ST5 (Figure 2). Data points were characterized bylocation on the plot as being from salt or sediment, and regions de-fined from which appropriate combinations of velocity and resistivityparameters could be drawn. The salt region was defined as a rectan-gular box, whereas the sediment relationship was defined by fitting apolynomial curve (equation 5). The fitted relationship and associateduncertainty for resistivity are shown in equations 5–7 and the equiv-alent relations for density are given in equations 8–9. The boundsshown here are for the 99% confidence bound (3σ). Data are kindlyprovided by Statoil.

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emulator was built independently for each modeling domain (seis-mic, gravity, and MT), using uniform distributions over the param-eters in columns 1–4 of Table 1, and without any statement aboutthe origin of the models being used.

Seismic emulator

In constructing a seismic emulator, we use a method similar tothat of Roberts et al. (2010, 2012), but here we develop and improvethe results by fitting weighting coefficients to Laguerre polynomialfunctions rather than fitting simple polynomial functions. Wechoose to use Laguerre polynomials because of their mutual ortho-gonality, which increases the efficiency in fitting the functions con-cerned. Other classes of orthogonal polynomials could have beenchosen here; however, Laguerre polynomials were a convenientchoice. The exponential weighting associated with Laguerre poly-nomials means that the fitting process here maybe more sensitive to lower parameter values. In amore thorough treatment, this could be a focusfor investigation; however, no significant issueswere encountered here, and they were deemed fitfor purpose. Depending on the setting, otherfunctions may be more suitable to choose for thebases; if the aim was to fit to periodic data, anatural choice of basis functions would havebeen a Fourier series, for example. For the firstcycle, we consider the velocity model spaceshown in Table 1, parameterized by 14 parame-ters; ðvm; smÞ7m¼1 where vi and si are the veloc-ities and thicknesses ascribed to each of the sevenlayers, as shown in Table 1. The model space isdesigned such that there is finer stratification inthe shallow region. This reflects the fact that as aresult of having traveltime data out to approxi-mately 10 km of offset, we expect greater seismicsensitivity in the upper 3 km or so. Our aim inbuilding the emulator is to predict, to a calibrateduncertainty, the seismic forward code output formodels drawn from this space. We generate a1500 × 14 Latin hypercube (McKay et al., 1979;Stein, 1987) and use this to create a set of 1500models over the 14-parameter space, which fillthe space evenly. Each of these 1500 14-param-eter models is then passed in turn to the forwardseismic simulator, producing 1500 t versus xplots, each consisting of 100 (x, t) pairs. The sim-ulator computes traveltimes using a finite elementmethod (Podvin and Lecomte, 1991; Heinckeet al., 2006). Laguerre polynomial functions arethen fitted, using a least-squares algorithm, to eachof these data sets (equation 10) to compute a vec-tor of polynomial coefficients αx;i to representeach of the i ¼ 1 − 1500 data sets. It was foundthat Laguerre polynomials of order 3, parameter-ized by four αx;i coefficients to weight the poly-nomials, are sufficient to recover the form of thedata and keep the least-squares algorithm stable.Our code is designed such that if a singularity oc-curs in the fitting of the coefficients (i.e., overfit-ting of the data is occurring), then the number of

coefficients is automatically decreased until a stable fit is achieved. Inearly versions of the code, simple polynomials were used as basisfunctions instead of Laguerre polynomials and overfitting of the datapoints was commonplace; however, using Laguerre polynomials,with the property of orthogonality over the space concerned, hasmeant that such overfitting using the number of coefficients specifiedhere has been eliminated. Thus, we reduce each plot of 100 datapoints to a set of four coefficients. In using these polynomial coef-ficients to represent the (x, t) data, there is a misfit function that wedenote as gxðxÞ, as follows:

t ¼�Xpx

i¼0

αi;xxie−xLiðxÞ�þ gxðxÞ: (10)

Table 1. Prior parameter bounds for each layer. Ranges are shown for emulatortraining, and the ranges used to sample models from each of the sediment andsalt cases. Velocity values are given in units of m∕s, resistivity values are givenin units of Ωm, density values are given in units of kg∕m, and the layerthickness values are given in units of m.

Layer ParameterTraining(min)

Training(max)

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Salt(min)

Salt(max)

1 Velocity 1600 5500 1600 5000 4000 5000

1 Resistivity 0.5 5000 0.5 10 100 5000

1 Density 1800 3600 1800 3600 2000 2200

1 Thickness 50 1600 50 1600 50 1600

2 Velocity 2000 5500 1600 5000 4000 5000

2 Resistivity 2.0 5000 2.0 20 100 5000

2 Density 1800 3600 1800 3600 2000 2200

2 Thickness 50 2700 50 2700 50 2700

3 Velocity 2000 6500 1600 5000 4000 5000

3 Resistivity 5.0 5000 5.0 70 100 5000

3 Density 1800 3600 1800 3600 2000 2200

3 Thickness 200 2900 200 2900 200 2900

4 Velocity 2000 6500 1600 5000 4000 5000

4 Resistivity 5.0 5000 5.0 70 100 5000

4 Density 1800 3600 1800 3600 2000 2200

4 Thickness 1200 2900 1200 2900 1200 2900

5 Velocity 2000 6500 1600 5000 4000 5000

5 Resistivity 5.0 5000 5.0 70 100 5000

5 Density 1800 3600 1800 3600 2000 2200

5 Thickness 1500 2500 1500 2500 1500 2500

6 Velocity 2000 6500 1600 5000 4000 5000

6 Resistivity 5.0 5000 5.0 70 100 5000

6 Density 1800 3600 1800 3600 2000 2200

6 Thickness 1500 2500 1500 2500 1500 2500

7 Velocity 2000 6500 1600 5000 4000 5000

7 Resistivity 5.0 5000 5.0 70 100 5000

7 Density 1800 3600 1800 3600 2000 2200

7 Thickness 1500 2500 1500 2500 1500 2500

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We then fit α4x;i¼1 to the model parameters (in this case, the veloc-

ity and layer thickness parameters ½vm; sm�7m¼1), again using a least-squares method to fit the weighting coefficients for Laguerre poly-nomials. This is similarly accomplished using Laguerre polynomialsup to third order in each of the layer parameters (equations 11 and12). The result is a set of 228 βx;ijk coefficients (four for each of the14 model parameters, plus a zeroth-order term, for each of the fourαx;i coefficients ð¼ ð4 × 14þ 1Þ × 4Þ). Again, there is a misfit func-tion associated with this fitting step (equation 13). Examples of therecovery of the αx coefficients using the βx coefficients are shownin Figure 4. Using these αx coefficients, we can then construct thetraveltime curves for a given set of model parameters. Examples com-paring the traveltime curves obtained using the recovered αx coeffi-cients with the simulated traveltime curves are shown in Figure 5:

θx ¼ ½v1 v2 v3 v4 v5 v6 v7 s1 s2 s3 s4 s5 s6 s7 �T(11)

and

αi;x ¼�Xwx

k¼1

Xqxj¼0

βijkθjk;xe

−θk;xLiðθk;xÞ�þ gi;xðθxÞ: (12)

In predicting the parametric components of the system, we havetwo sources of misfit in the process of building the emulator: gxðxÞand gx;iðθxÞ, as in equations 10 and 12, respectively. In equations 12–15, we group the terms so as to separate the parametric and nonpara-metric parts of the system and obtain the global misfit functionGðx; θxÞ, which is a function of offset x and the model parametersθx. A more careful treatment of the systemwould involve consideringthis dependence. However, on the grounds of simplicity of calibrationgiven the proof-of-concept nature of this study, we chose to computea misfit function averaged over all model parameters. Thus, we con-sider the misfit function GxðxÞ, as shown in equations 16 and 17:

t ¼Xpx

i¼0

Xwx

k¼1

Xqxj¼0

βijkθjk;xe

−θk;xLiðθk;xÞxie−xLiðxÞ

þ gi;xðθxÞxie−xLiðxÞ þ gxðxÞ; (13)

¼Xpx

i¼0

Xwx

k¼1

Xqxj¼0

βijkθjk;xe

−θk;xLiðθk;xÞxie−xLiðxÞ

þ�Xpx

i¼0

ðgx;iðθxÞxie−xLiðxÞÞ þ gxðxÞ�; (14)

Seismic

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Figure 4. Example reconstruction of α coefficients from β coefficients for ST12. In the case of the gravity emulator, note that we are simplyrepresenting a single point, rather than a function represented by α coefficients, and so we plot the emulator-reconstructed points against the“actual” points generated by the gravity simulator. Note the strong correlation between emulated and simulated outputs in each case.

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¼Xpx

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−θk;xLiðθk;xÞxie−xLiðxÞ þ Gðx; θxÞ;

(15)

≈Xpx

i¼0

Xwx

k¼1

Xqxj¼0

βijkθjk;xe

−θk;xLiðθk;xÞxie−xLiðxÞ þGxðxÞ;

(16)

and

GxðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnmax

n¼1 ðtem;nðxnÞ − tsim;nðxnÞÞ2nmax

s: (17)

To properly effect the screening of model space using a predictivesimulator proxy, it is necessary to calibrate the uncertainty on thepredictor. We calibrate the emulator uncertainty in using βx for pre-diction by computing GxðxÞ, as in equation A-8. This is done bygenerating the βx-coefficient estimated output function and the fullsimulator output for each training model parameter set (Figure 5)and computing the root mean square (rms) of the residuals withrespect to the traveltime functions used to train the emulator as afunction of x. Examples of this misfit function are shown inFigures 6a–9a.A key question is, “How many training models are required to

correctly estimate GxðxÞ and thus sufficiently sample the modelspace?” This question is vitally important for two reasons: first, be-cause a model’s plausibility or implausibility can only be reliablydetermined if the emulator uncertainty with respect to the estimationof the simulator output is correct and second because the aim at eachscreening cycle is to exclude model space not deemed plausible; it iscrucial to properly sample the whole of the remaining space to pre-vent the model space from being wrongly removed. If the modelcoverage is not sufficient, then the emulator will underestimate thepredictive uncertainty. A more rigorous study would involve either amore detailed assessment of the space to be sampled or the inclusionin the sampling method of a finite probability of sampling outsidethe currently constrained space. Using the criterion of two samples/parameter, we would wish to use 214 ≈ 16000 training models (forexample, as in Sambridge and Mosegaard, 2002). For our purposes,we chose a semiqualitative and fairly rudimentary approach of con-sidering that if the coverage is sufficient, then the addition of furthermodel parameter sets to the training process will not significantlyalter the uncertainty estimate. We therefore calibrated the number ofmodels needed by testing cases of generating the emulator using150, 1500, and 15,000 training models and assessing the impacton the emulator uncertainty of adding more models to the trainingprocess. For emulators trained over our prior model space (Table 1),it was found that using 150 models was insufficient (Figure 5), butthat the uncertainty function estimates using 1500 and 15,000 train-ing models give similar uncertainty functions. Over this space,therefore, 1500 models is deemed a sufficient number with whichto train the emulator.The set of βx coefficients and this uncertainty function GxðxÞ

together constitute the emulator, or statistical model. We use this

uncertainty function to determine whether emulated output dataof a proposed model lie sufficiently close to the observed data setssuch that the model can be deemed plausible or not. However,GxðxÞ is calculated as the rms of the simulator-predictor residual,and as such it is possible (and indeed, it is certainly the case in someinstances) that the actual data-representation error for a given set ofmodel parameters may be significantly larger than this. Hence, itmay be the case that potentially plausible models are rejected by theemulator screening simply because the emulator prediction for thatset of model parameters was located in the tail of the uncertaintyfunction. The emulator screening reliability is, therefore, tested byusing this screening technique on 100 target data sets, produced bythe simulator from 100 synthetic models. A scaling factor γx for theuncertainty function is then calculated by calibrating against these100 target data sets, such that there is at least a 97% probability thatthe emulator screening process will include the “true” model in itsselection of plausible models if the true model is included in thecandidate model space. The figure of 97% is in many senses arbi-trary; however, we considered it suitable for the purpose at hand.The condition for plausibility is shown in equation 18, where

2 4 6 8 100

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a)

b)

Figure 5. (a) Four example traveltime training outputs and emulator-reconstructed outputs. Black ovals show the traveltimes generated bythe full simulator code, and the gray lines show the traveltime curvespredicted by applying the predictive β coefficients to the same sets ofmodel parameters. (b) Comparison of seismic emulator uncertaintyfunction for ST13 after eight cycles using 15,000, 1500, and 150training models.

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temðxnÞ and ttargðxnÞ are the emulated and full simulator traveltimesat an offset xn, respectively. The weights κx;n are user-definedweights for each traveltime point. For example, we attach greaterimportance to achieving a close fit to the short-offset traveltimes,compared with the long-offset measurements on the basis that thevelocity gradient is typically higher in the shallow structure. Table 4shows the values of κ used in this study. Here, we have chosen togive all points a weighting of either 1 or 0, and varied the density ofpoints along the offset profile with value 1 to control the weightgiven to varying parts of the traveltime curves. If preferred, the usercould easily use fractional weights:

Xnmax

n¼1

κx;nmax½jðtemðxiÞ − ttargðxiÞÞj − γxGxðxiÞ; 0�

GxðxnÞPnmax

p¼1 κx;p< nmax:

(18)

Spike emulator

To locate discontinuities in the gradient of the seismic traveltimecurves and thus constrain abrupt changes in velocity at layer boun-daries, a “spike” emulator was built. There are a number of otherapproaches (Grady and Polimeni, 2010), which could have beentaken to identify the boundary positions, such as the basic energy

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ST5

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Figure 6. (a) Seismic, (b) MT, and (c) gravity uncertainty functionsfor ST5. Arrows show how the predictive uncertainty of the emu-lators reduce with the increasing screening cycle as the model spaceis refined.

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model and the total variation model. Each has strengths and weak-nesses, particularly regarding how noise is regarded in associationwith high-frequency data. A key part of the philosophy of our meth-od is that it should be as conceptually straightforward as possibleand data driven wherever appropriate, and so we chose to imple-ment the simple gradient detection method, described here. In build-ing the seismic emulator, the chosen form of data reduction of usingpolynomial curves to represent the t versus x curves, while beingsuitable for describing the smooth trends (Figure 5), does not cap-ture discontinuities in the traveltime gradient function dt∕dx. Rob-erts et al. (2012) describe a strategy to consider these gradient

discontinuities, whereby the dependence of the offset position ofthese gradient discontinuities is considered as a function of the seis-mic model parameters. We adopt the same strategy here, seeking tomodel such features in the data, and thus capture structural infor-mation, with a view to optimizing the positions of the model layerboundaries, thus best representing the substructure.As in Roberts et al., (2012), instead of considering dt∕dx to probe

this information, we calculate the squared second derivative of the tversus x function ψ ¼ ðd2t∕dx2Þ2 (Figure 10). In principle, giventhat we are using seven-layer models, to optimize the layer boun-dary positions, we could search for the six largest spikes. However,the presence of six discernible spikes in many of the observedseismic CMP gathers is unlikely (see Figure 2 for example), andthis may yield the positions of noise spikes (the positions of whichwould likely be uncorrelated to any structural information). Toavoid potential computational problems as a result of misattributingstructurally sourced gradient discontinuities to noise, we choose toonly estimate the offset positions x of the three largest spikes in thisψ ¼ ðd2t∕dx2Þ2 function. We preferentially use ðd2t∕dx2Þ2 as op-posed to d2t∕dx2 to ensure that ψ is positive, simplifying the proc-ess of picking the extrema, in addition to exaggerating the relativemagnitudes of the spikes in question. A key assumption of thismethod is that the largest spikes do represent layer boundaries,rather than noise. For cases in which there is a high degree of noise,it may be necessary to either consider other methods for the detec-tion of structural boundaries or reduce the number of model layersand the expected output resolution.For each seismic emulator training data set, we therefore compute

(numerically) ψ ¼ ðd2t∕dx2Þ2 and then search for the offset x

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ST1 ST2 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST10 ST11ST12 ST13 ST14

ST1 ST2 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST10 ST11ST12 ST13 ST14

Figure 9. (a) Seismic and (b-c) MT emulator uncertainty functions(GxðxÞ and GωðωÞ) for each station. These data maps represent thepredictive uncertainty of the emulator in predicting the simulator out-put. White lines show the positions of the stations between which thefunction is interpolated.

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positions of the three largest spikes. These are then fitted to themodel parameters ðvm; smÞ7m¼1 according to the formulation inequations A-9–A-12 and the predictive uncertainty Gψ ;i, computedby comparing the “actual” positions of the spikes to the estimatesgiven by βψ.

MT and gravity emulators

We built emulators for each of the MT and gravity data sets in ananalogs way; to predict complex impedance and gravitational fieldstrength as a function of the resistivity and density models, respec-tively (Roberts et al., 2012). For the MT case, unlike with the seismicdata, the (ω, ReðZÞ) and (ω, ImðZÞ) functions are smooth functions,and so we do not construct an analog for the seismic spike emulatorfor the MT data set. In the case of the gravity emulator, because thereis simply a single gravity measurement at each location, rather than afunction such as (x, t) or (ω, ReðZÞ) or (ω, ImðZÞ), the initial datareduction step is not necessary, and so we simply fit the output sim-ulator gravity value to polynomials in the model parameters, in a sim-ilar fashion to the method used to construct the spike emulator.

Screening phase

Having trained an emulator for each of the seismic, spike, gravity,and MT cases using the method described above over models gen-erated from the parameter ranges in Table 1, we generate candidate

models to test for implausibility across the three data sets: seismic,gravity, and MT. Our goal is to generate candidate joint models totest for implausibility and thus discern the commonly plausible setof models (as illustrated in Figure 11). To maximize the constraintafforded by the process, rather than performing a single screeningphase, we repeat the screening process in a cyclic scheme, each timeusing the remaining plausible model space from the previousscreening cycle to build a new emulator, which due to being trainedover a smaller space, will have a smaller GxðxÞ and thus be morepredictively accurate (see Roberts et al. [2010] and Figures 6a–9).For each of the seismic, spike, gravity, and MT cases, we use the

respective emulator to rapidly test sets of model parameters to see ifthe emulated output from each set of model parameters lies within agiven range fγqGqgq¼x;ψ ;ω;ρ of the observed wide-angle traveltimedata, gravity measurement, and ReðZÞ and ImðZÞ data observedclosest to each of the stations ST1-14.The generation of joint models may be accomplished in a

variety of ways. We choose to generate such 28-parameter modelsðvm; rm; ρm; smÞ7m¼1 by choosing a set of (vm, sm). Resistivity anddensity values (rm and ρm) are then generated using the appropriate(salt or nonsalt) relationship from vm according to equations 5–9. Ingenerating joint candidate models; therefore, our first question foreach layer of the model we are generating is whether we wish thatlayer to be characterized by a salt relationship or a sediment rela-tionship (Figure 3). At the start of our analysis, we therefore specifya probability for each layer of the model, pm with which to generatecandidate models using the salt relationship or the sediment rela-tionship. Here, we set this probability to 0.5 for each layer. Inthe first screening cycle, ðvm; smÞ7m¼1 are generated on the fly usinga Sobol algorithm (Bratley and Fox, 1988), and then the emulatoroutputs are tested for plausibility against the observed data set atthe station in question. In using common layer thicknesses forthe models, we are imposing the additional constraint of structuralcoherency across the models. We then generate (rm, ρm) from vmaccording to the salt/sediment probability vector pm. Note that indoing so, the prior probability distributions for resistivity and den-sity specified in Table 1 become largely implicit in the screeningprocess; however, in the initial emulator training step, models foreach domain are drawn independently of the coupling relationshipsin equations 5–9.On generating each candidate model, we use each of the four

emulators to generate an estimated data output in each case. Wedefine the condition for implausibility for an individual methodas follows: A weighted mean of the emulator-predicted data resid-uals with respect to the observed data is less than γxGxðxÞ, γψGψ ,γωGωðωÞ, or γρGρ as appropriate. A joint model is considered com-monly plausible, and thus suitable for use in the subsequent cycle, ifit is found to be not implausible with respect to all three methods:seismic (including spike), MT, and gravity, based on the plausibilityconditions shown in equations A-31–A-35 (Figure 11). By gener-ating and testing model parameter sets in this way, we reject theimplausible model space and we build up a population of plausiblemodels. This is then repeated in a cyclic fashion; when 1500plausible models have been found, they are used to build a newemulator, which is used alongside the previous emulators to screenfurther models from the reduced model space. A more careful treat-ment would calibrate the number of training runs for each cycle(by determining when the addition of further training runs has anegligible impact on the uncertainty estimate).

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Figure 11. Commonly plausible model statistics using the seismic,gravity, and MT emulators to generate a population of 1500 plau-sible models for one screening cycle at station ST12. In each case,the numbers show the number of models deemed plausible by eachscreening method: seismic (including spike), gravity, and MT.

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Note that for cycle n, where n > 1, the condition for a candidatemodel to be considered plausible is not simply that it is not consid-ered implausible by the test described using the emulators con-structed for cycle n using the plausible models from cycle n − 1,but that it is also not considered implausible for each of the preced-ing cycles. Thus, for a given candidate model at cycle n, it must passthe screening test for 4n sets of emulated data outputs (one for eachof the n cycles, over each of the four methods [seismic, spike, grav-ity, and MT]). However, as soon as a model is deemed implausibleby a single screening, then that candidate model can be discardedand a new candidate model can be generated. To maximize screen-ing efficiency, we choose to compute the emulated output for agiven model for whichever of the emulators (seismic, spike, gravity,or MT) is fastest to run, including computation of the plausibilitycondition (equations A-31–A-35). In this case, testing against thegravity measurement was fastest because it is just a single point.After the first cycle, to maximize efficiency, rather than using the

Sobol sampling strategy, we use a method to sample from the jointmodel parameter distribution. Sampling from the joint distributionis not a trivial task, and there are a number of ways of accomplish-ing this, for example, the method of Osypov et al. (2011). One of thekey considerations is that although we have 1500 models, whichsufficiently sample the model space, we do not wish to sample ina bootstrap manner from this distribution of point values, but fromthe continuous distribution described by these points. If we were torepeatedly sample simply from the distribution of point values fromthe previous cycle, then after a few cycles, the resulting distributionwill tend toward that of a number of discrete spikes. To avoid thisissue, we use a scheme of sampling the combinations of velocityand thickness parameters (vm;i and sm;i) from the previous cycle,each perturbed by a value sampled from a uniform distribution witha width of 1% of the marginal plausible parameter range from theprevious cycle. In this manner, we generate new sets of ðvm; smÞ7m¼1

values which are close to those deemed plausible in the previouscycle, according to the formulation in equation A-36. The pertur-bation of 1% was chosen after testing a range of values. The greaterthe value that is chosen, the greater is the continuity of the overalldistribution, at the expense of smearing the information available.We make the perturbation using a uniform distribution rather than anormal distribution to avoid issues relating to leakage, particularlywhen sampling from close to the bounds of the previous cycleparameter distributions on the grounds that Uða; bÞ is bounded be-tween a and b, whereas Nðx̄; σÞ is unbounded. Other distributions,such as a β-distribution, could be used here, and they may be con-sidered to be better choices; however, in this case, we chose to use auniform distribution for conceptual and computational simplicity.The choice of perturbation method affects how the distributionsare sampled, and thus they are something upon which the final re-sults are dependent, and so they should be given proper thought.The total number of screening cycles used can be determined by

one of several methods: that the size of the emulator uncertaintyfunctions fGqgq¼x;ψ ;ω;ρ fall below some threshold value, or thatthey cease to reduce further (at which point, all discernible paramet-ric information, given our emulator parameterization, has been ex-tracted from the system), or at some arbitrary fixed number. In ourcase, we choose to use a fixed number of 25 cycles. We considerthat setting an arbitrary cutoff in this fashion is not the most rigorousmethod, and for a robustly interpretable result, this should begiven greater consideration; however, we deemed it suitable for

the purpose at hand of demonstrating the screening methodology.The result from a screening cycle is the joint distribution of modelsnot deemed implausible. Figure 12 shows example marginal distri-butions of model parameters for the plausible models after 25 screen-ing cycles for station ST5. These do not represent the full informationavailable from the joint distribution; however, they are useful inunderstanding how the parameter space is being constrained. Weconsider it to be of much greater interpretative value, however, toconsider the acceptance ratio plots of Figures 13–16 (after Flechaet al., 2013), which show the prior model space and the acceptanceratio for model parameters in parameter-depth space.After each screening cycle, the population of plausible models is

analyzed to ascertain the proportion of models that are characterizedby salt or sediment for each layer (this can be considered an extramodel parameter). This proportion is then used to update the prob-ability vector p7

m¼1 during the subsequent screening cycle that a par-ticular layer in each candidate model will be generated using the saltor sediment relationship (equations 5–9). The parameter bounds aresimilarly updated based on the newly refined model space.By averaging, or computing particular parameter quantiles over

the depth range, and interpolating between the station, parametermaps such as Figures 17–19 can be generated. These plots are

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useful for gaining a broad overview of the geophysical featuresalong the profile; however, particularly in cases where there isstrong multimodality in the plausible model parameter distributions(as is seen in several cases in Figures 13–16), they can be stronglymisleading in that in such cases the average model may lie close to aminimum in acceptance ratio, rather than representing a maximumlikelihood estimate. This is considered further in the “Discussion”section.As well as the ability to consider the distribution of parameter

likelihood, this method also allows the construction of a rock prob-ability map, as in Figure 20. This is constructed by calculating thefraction of salt and sediment physical parameter relationships thatwere deemed plausible at depth nodes down the profile.

Model validation

Well-log data

Figure 16 shows log data from a well located close to station ST5,the location of which is shown in Figure 2. Note that, as can be seenfrom Figures 17–19, the well track is not vertical at the location ofST5 (in fact, it is much closer to ST6 at depth), and so this overlay isonly semiapplicable, and it should not be used for detailed compari-son purposes. However, it can be seen from Figures 16–19 that thescreening process has done a reasonable job in seeing the structuralvariation observed in the borehole.

In the case of resistivity, the well-log plot is close to the upper endof the 10%–90% band through most of the depth range and themedian quantile is far from the well-log plot. Note that these quan-tiles are not used for any screening purpose, but they are shown forillustrative and visualization purposes. The presence of the “groundtruth” well-log plot toward the edge of the plausible model spacefurther emphasizes that simply adopting some maximum likelihood,mean, or median model can be misleading, and that simply adoptinga median/mean/modal model as being representative of the modelspace would not be appropriate in this case.The density log overlay of density profile shown in Figures 16

and 17–19 is less useful because the log only starts near the top ofthe salt body. However, within the joint setting of the screeningprocess, the method correctly identifies the top of the salt bodystructure. Note that the density log is best described by the lowerdensity quantile shown in Figure 19c, reflecting the likely presenceof salt.

Deterministic inversion

Coworkers on the Joint Inversion with Bayesian Analysis (JIBA)project carried out a 3D deterministic inversion of the same jointdata set using the method of Moorkamp et al. (2011, 2013). Theirresult for the velocity profile is shown in Figure 21. For the MT andgravity results associated with the deterministic inversion, the

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reader is referred to Moorkamp et al. (2011). Comparing this withFigures 17a and 20, it is seen that the salt body is collocated withthat discerned through the model screening method.

DISCUSSION

Emulator versus simulator run time

A key aim of the emulator screening method is that the use of anemulator should afford a considerable time savings over the full sim-ulator code for the computation of outputs for a given set of modelparameters. To quantify this, an emulator construction and screeningcycle was timed for one of the stations (ST14) for the seismic sim-ulator code (the most complex to fit). To run the 1500 training modelstook 353 s with the full simulator. In the subsequent screening cycle,11,243 models were then screened using the seismic emulator in 22 sto obtain 1500 plausible models for the next cycle. The number ofmodels computed per second by the simulator and emulator are thusthose given in Table 2.In this particular case, therefore, the emulator can screen models

around 100 times more rapidly than using the full simulator. Thisillustrates the value of using an emulator in this setting, and indeedalthough the probem at hand in this case is relatively simple, the morecomplex the simulator code is, the greater will be the time saving thata well-designed emulator can produce.

Recovery of α coefficients

In general, the recovery of the coefficients (Figure 4) for the seis-mic emulator is reasonable and there is a clear correlation betweenthe “real” αx coefficients obtained from fitting curves to the trainingdata sets, and the emulator-reconstructed αx;em, obtained by usingthe predictive β coefficients with the same model parameter sets.The gravity data points are also well reconstructed. There is aslightly higher scatter on the MT coefficient recovery plots, mean-ing that the MT emulator appears to be slightly less effective at pre-dicting the form of the output data for a given input model; this doesnot mean that the MT screening is less reliable. This is because theuncertainty in prediction is absorbed by a larger uncertainty func-tion. Hence, although the rate at which the emulator can excludeplausible model space is lower, the reliability of the screening proc-ess itself is not affected. The function of the emulator is not to justrapidly predict the simulator output for a given model, but to do sowithin a calibrated/known uncertainty. This highlights how concep-tually different the emulation Monte Carlo approach is to many of thecurrent schemes, which seek to simply model the system as accu-rately as possible and find the best model. Here, instead, we seekto iteratively exclude implausible model space, until further exclusionis not possible, and the ability to do this relies not only on the abilityto predict the data for a given model, but also on knowing the

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uncertainty of that prediction. The reliability of the emulator in doingthis is in fact thus determined as described earlier, by the scaling fac-tor fγqgq¼x;ψ ;ω;ρ, which is applied to the uncertainty function

fGqðqÞgq¼x;ψ ;ω;ρ, which is calibrated when the emulator is built.

Spike emulator methodology

In modeling the positions of discontinuities in the seismic trav-eltime gradient function, one approach could have been to fit a pol-ynomial function to the derivative dt∕dx. However, we chose toadopt the described approach of fitting the offset positions of thespikes in the ψ ¼ ðd2t∕dx2Þ2 function. This latter approach wasfavored because, given we are using a polynomial to representthe t versus x function, if we try to fit a polynomial to the derivativeof this function, dt∕dx, the result of the least-squares fit is likely tobe the derivative of the function given by our α-coefficient polyno-mial representation, which we could calculate analytically, and sowe would not gain further useful information. In addition, the parts

of the gradient function containing the most useful information arethe steepest-turning regions, “which are the most difficult parts to fitusing smooth functions. Another advantage of the spike-fitting ap-proach over trying to predict the gradient function itself is that themaximum number of data points we are aiming to fit for an n-lay-ered model is n–1. Whether we choose to fit all n–1 points in thisway or as described in the “Methodology” section, in our case, onlythree data points (the x-positions of the three largest spikes in theψ ¼ ðd2t∕dx2Þ2 function), the emulator screening process is consid-erably more efficient than in the case of fitting the derivative func-tion dt∕dx to the entire set of traveltime offsets.

Emulator uncertainty reduction and model spacerejection rate

We have seismic data with traveltime offsets to 10 km, so weexpect (by rule of thumb) a seismic resolution down to approxi-mately a 3 km depth. In designing the seven-layer space, we there-

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Figure 17. (a) Mean, (b) upper, and (c) lower 90% quantile velocitymodels. Generated by calculating the distribution of velocity param-eters at depth nodes for each station and interpolating between sta-tions. The borehole track is overlaid with colors indicating the logvelocity. Note that the log velocity is very similar to the upper veloc-ity quantile in panel (b), consistent with the presence of salt in thatregion.

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Figure 18. (a) Mean, (b) upper, and (c) lower 90% quantile resis-tivity models. Generated by calculating the distribution of resistivityparameters at depth nodes for each station and interpolating be-tween stations. The borehole track is overlaid with colors indicatingthe log resistivity. Note that the log resistivity is very similar to theupper resistivity quantile in panel (b), consistent with the presenceof salt in that region.

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fore concentrated thin layers toward the upper 1 km of the models(due to the nature of the method, we expect the seismic data to af-ford the greatest constraint at shallow depths in comparison to theMT and gravity methods). This can be seen in Table 1 by looking atthe permitted layer thickness ranges. In the histograms of Figure 12,the prior parameter ranges are represented by the horizontal axiswidths of each histogram. These correspond to the ranges shownin Table 1. The fact that the distribution widths of the final plausibleparameter ranges are smaller than these ranges in most cases dem-onstrates how the system is constraining the plausible model space.This is seen even more strikingly in the acceptance ratio plots (Fig-ures 13–16), where the black background shows the prior modelspace. This model space reduction is also reflected in the uncer-tainty functions in Figures 6–8, which reduce at each new cycle.The seismic and MT uncertainty functions at the end of the finalscreening cycle for each station location are shown in the formof predictive data uncertainty maps in Figure 9.Figures 6–8 show how the predictive uncertainty of the emulator

in representing the simulator output reduces with each cycle as themodel space is refined. It is clear that the rate of this reduction (alsovisible in Figure 9) varies across the profile: The rate of uncertaintyreduction at station ST7 is much lower than at station ST13, for ex-ample. This variation is directly related to the changing plausiblemodel parameter space, and the nature of the interparameter relation-ships in equations 5–9 and Figure 3; a high uncertainty in these willreduce the rate at which model space is rejected as implausible. In thiscase, there is a large uncertainty on the resistivity of salt (Figure 3) incomparison with the uncertainty for sediment. Also in the case of salt,there is no correlation between the resistivity, density, and velocityvalues in that the value of resistivity does not further constrain thevalue of velocity or density. This can be seen in Figure 3, in whichthe relationship bounds for salt are simply rectangular boxes, as op-posed to the nonzero gradient on the relationships for sediment.As a result of these two considerations, in regions where there is

a strong salt presence, the system will be comparatively slow inrejecting the model space because the three methods (seismic, grav-ity, andMT) are not strongly mutually cooperative in reducing modelspace. In the case of sediment, there is lower uncertainty in the rela-tionship and a much stronger correlation between the parameter val-ues. As a result, where there is sediment, the three methods assist oneanother much more effectively in reducing the size of the modelspace. The cooperation between the methods can be seen in Figure 8,in which each of the methods is seen to be clearly more effective inreducing the emulator predictive uncertainty at varying screeningcycles. This illustrates the value of a joint approach over a singleparameter analysis in that each method contributes information thatcan be used to constrain the parameter values for the other methods.The results of Moorkamp et al. (2011, 2013) demonstrate this pointwell. In regions where there is salt, the emulator uncertainty reductionrate is much lower (Figure 8), the result of which is that in suchregions, sediment models will be rejected less easily. The effect ofthis is that where salt is present, the probability of salt presence(Figure 20) will be underestimated, or rather, it will be biased toward0.5, which is the prior specified salt probability across the model. Theresult is that, due to the nature of each of the physical parameter cou-pling relationships, the system will more easily discern the presenceof sediment than the presence of salt. This is reflected in Figure 20,where the system seems more confident of the suitability of the sedi-ment relationship being appropriate in regions of sediment. As was

commented earlier regarding Figure 3, there is a case for perhapsincluding a third rock-type relationship in the analysis given thereare a number of observations lying outside the confidence rangesof the fitted relationships.

Nonmarginal information and sampling strategies

The histograms of Figure 12, however, do not show the full extentby which the parameter space has been shrunk. This is because theysimply show the distribution of marginal model parameters for theplausible model space at each cycle. To maximize the efficiencyof the sampling scheme, rather than simply sampling parametersfrom the univariate marginal distributions of each of the layers, asis the case here, the scientist could sample from the full joint distri-bution of parameters across all layers. This would mean that, for ex-ample, emergent correlations between, for example, the velocity oflayer 1 and that of layer 2, could be used. Such a scheme is discussedfor a synthetic case in Roberts et al. (2010, 2012).Although the principle of sampling from a prior model space and

testing models for plausibility is conceptually straightforward, themanner in which this sampling is carried out (in particular the priordistribution of parameters) requires some careful thought in eachcase. This is because the shape of this distribution (whether it benormal, uniform, or some other class) is, itself a positive prior state-ment of belief about the system. It is thus of considerable impor-tance that the sampling strategy and prior parameter distributionsare given careful thought before embarking on this kind of method.Sampling issues — One fundamental weakness of the method as

currently used is the assumption that the plausible model space for agiven emulation cycle has been sufficiently sampled when 1500 suc-cessful models have been found. A proper treatment of the problemwould include the consideration that at any given emulation cycle,there is the possibility that plausible areas of model space have notbeen sampled. A significant improvement to this proof-of-conceptmethodology would thus include, at each emulation cycle, a finiteprobability of sampling models outside the currently constrainedplausible model space (a “jumping” distribution), as is commonlyimplemented in Metropolis-Hastings-based sampling schemes(Metropolis et al., 1953; Hastings, 1970).Given the nature of the screening-cycling method in which the

first n successful models are chosen for screening in the next cycle,it should be borne in mind that a weakness of this screening schemeis that over time, for bimodal distribution sampling, such as is thecase here where salt and sediment populations are being sampled,there will be bias toward asymmetry when the local maxima arenot equal.In addition to the prior model space and coupling parameters

(Table 1 and equations 5–9), the method presented here also in-cludes a number of tuning parameters as inputs (Figure 1), such asthe number of models used for each screening cycle (1500) or theperturbation of 1% in sampling model parameters from the previouscycle. As we have presented it here, these tuning parameters havebeen arbitrarily selected, rather than fully sampled. Because the re-sults are affected by the values chosen for these tuning parameters,although they are probabilistic and the probability map shown inFigure 20 is not a truly Bayesian result in that it does not representthe product of the prior salt probability and a likelihood function. Avaluable further development of this methodology would be to con-sider the sampling of these parameters more rigorously.

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Model representation

One of the key advantages of this kind of approach is that, ratherthan a single average model, and perhaps an estimate of the uncer-tainty on it, here, the result is the plausible model space. Although theinformation contained in this result is considerably richer than in thecase of an average model and an uncertainty estimate, representingthis information presents a challenge. Figures 13–15 (after those ofFlecha et al., 2013) show the acceptance ratio for model parameterswith depth at a number of stations. It is quickly seen that, while insome cases (e.g., station ST13), there is a clear, well-defined unim-odal parameter distribution over most of the depth range, in othercases (particularly where it is thought that there is a considerable saltpresence, e.g., stations ST7–8), the distribution of plausible param-eters is multimodal. This reflects the fact that, given the observed dataand the specified parameter relationships, there may be either sedi-ment or salt present at given depths. This multimodality shows how itcan often be inappropriate to represent the resulting geophysical

parameter constraint as some uncertainty around a central averagevalue. In recent years, however, a few authors, such as Zhdanov et al.(2012), have made progress in developing methods for joint inversionschemes for multimodal parameter spaces.This is further highlighted by Figures 16–19. From Figure 16a

and 16c, we see that over much more of the depth range, thereare two distinct populations of parameters accepted by the screeningprocess; in the case of velocity, the higher valued velocities are fromsalt models, and the lower valued velocities are from sediment mod-els. In the case of density, the reverse is the case (compare withFigure 3). In this case, it is clear that representing the result by acentral average value, where the acceptance ratio is zero, or negli-gible, would be highly misleading. In this case, it is both of the ex-trema of the accepted parameters that more appropriately representthe result. Figures 17 and 19 demonstrate this clearly, in that the well-log velocity and density re much better represented by the 90% and10% quantiles, respectively, as opposed to the means in each case.Note that in the case of resistivity (Figures 16b and 18), there is a

more unimodal output over the depth range. It can also be seen thatthe system consistently accepts resistivity values lower than does therecorded well log. This is thought to be due to the fact that the MT

Dep

th (

km)

0

1

2

5

4

3

0 2000 4000 6000 8000 10000

Distance (m)

0.6

0.4

0

0.2

0.8

P(salt)1.0

Dep

th (

km)

0

1

2

5

4

3

0 2000 4000 6000 8000 10000Distance (m)

0.10

0.05

0.00ST1 ST2 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST10 ST14ST11 ST12 ST13

(P(salt))0.15

ST1 ST2 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST10 ST14ST11 ST12 ST13

σ

a)

b)

Figure 20. (a) Salt probability map: generated by averaging themodel count for “salt” models deemed plausible at depth nodes foreach station, and interpolating between stations. (b) Standard devia-tion of the probability estimate.

0

2

4

Dep

th (

km)

0 8000 10000Distance (m)

2000

3000

4000

5000

Velocity(m/s)

ST1 ST2 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST10 ST14ST11 ST12 ST13

1

3

5

2000 4000 6000

Figure 21. Velocity model result from the joint inversion carriedout by Moorkamp et al. (2011, 2013). The borehole track is overlaidwith colors indicating the log velocity.

b)

a)

c)

Dep

th (

km)

0

1

2

5

4

3

0 2000 4000 6000 8000 10000

Density: 90% (upper) quantile

Dep

th (

km)

0

1

2

5

4

3

0 2000 4000 6000 8000 10000

23002100 2200 260025002400

Density (kgm–3)2000

Density: mean

Dep

th (

km)

0

1

2

5

4

3

0 2000 4000 6000 8000 10000

Density: 10% (lower) quantile

Distance (m)

ST1 ST2 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST10 ST11ST12 ST13 ST14

ST1 ST2 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST10 ST11ST12 ST13 ST14

ST1 ST2 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST10 ST11ST12 ST13 ST14

Figure 19. (a) Mean, (b) upper, and (c) lower 90% quantile densitymodels. Generated by calculating the distribution of density param-eters at depth nodes for each station and interpolating between sta-tions. The borehole track is overlaid with colors indicating the logdensity. Note that the log density is very similar to the lower densityquantile in panel (c), consistent with the presence of salt in that re-gion.

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recording instruments are sensitive to conductivity rather than resis-tivity, so that layers of high conductivity/low resistivity are seen withgreater resolution than low-conductivity/high-resistivity layers.

Beyond parameter models: A rock probability map

Amajor benefit of this model sampling approach is that, from thestart, the question of discerning between two types of rock (de-scribed by the parameter relationships in equations 5–9) has beenintegral. Rather than simply seeking to test model parameters, wehave tested joint models consistent with either one parameter rela-tionship or another (in this case, sediment or salt). This allows theacceptance ratio for each relationship to be considered and a map ofthis, as well as of the acceptance spread, to be constructed (Fig-ure 20). This result should be understood in context, in that it givesthe model acceptance fraction given all of the prior model spacespecifications and tuning parameters used in this study. As has beencommented in the methodological sections, here, the model spaceand tuning parameters have been selected somewhat illustratively,with the aim being to focus on the screeningmethod. The result presented here should there-fore not be understood as a robust estimation forthe region in question. For the result to be geo-logically meaningful and robust, suitable expertconsideration should be given to the prior modelspace and tuning parameters.From Figure 20, we immediately see how the

system has much more definitely discerned thepresence of sediment (indicated by blue colorsand acceptance ratios < 0.5) than the presenceof salt (red colors and acceptance ratios >0.5).We consider this to be due to the nature of thephysical parameter relationships (Figures 3),and it is something worth future investigation.We see that the central region (ST6–9) has amuch lower likelihood of sediment presence inthe shallow subsurface in comparison to the restof the profile. It is very striking, on comparingFigure 20 with Figures 13–19, that the regionin which there is apparently more salt is not nec-essarily characterized by a well-defined param-eter model (velocity, resistivity, or density). Ifthe uncertainty shown in Figures 13–19 werepresented as the result of an inversion, it wouldlikely be considered that the data were at fault, orthat (rightly) it was not possible to constrain theregion from the observed data. However, on con-sidering Figure 3, we see that it is this very un-certainty in the parameter map, which reflects thefact that salt is more likely to be present. How-ever, as has been noted, the peak acceptance ratiois approximately 0.5, which is the same asthe prior probability specification, suggestingthat the system has also struggled to add dis-cernment.

Large 3D models and data sets

The study presented here may be consideredrelatively simplistic given the size of typical 3D

models and data sets, and it is a pertinent question as to how thismethodology may be scaled up to a fully 3D treatment of the prob-lem at hand. We have shown how using emulators instead of the fullsimulators can increase greatly the efficiency of screening a largemodel space. However, the practical usefulness of this method de-pends on the time required to build the emulators. For large modelspaces with large numbers of parameters, even running the numberof simulations required to build an emulator may be impractical. In

Table 2. Model computation rate for the seismic simulatorversus emulator for one screening cycle.

Data set realizations/second

Simulator 4.25

Emulator 511.04

Table 3. Mathematical symbols used in this paper.

Symbol Description

n Data point reference number (for seismic, gravity, or MT)

nmax Total number of data points (for seismic, gravity, or MT)

x, xn Offset at which traveltime t is observed

t, tn Traveltime (at data point reference n)

ω, ωn Frequency points at which MT impedance Z is observed

Z, Zn MT impedance measurements

R, I Real and imaginary parts of Z

ψ , ψn Squared second-derivative of the traveltime curve

ϕ Gravity measurements

θ Model parameters (v, r, ρ, s)

v, vm P-wave velocity of layer m

ρ, ρm Density of layer m

r, rm Resistivity of layer m

s, sm Thickness of layer m

p, pm Probability that layer m is “salt”α, αi, αq;i Coefficients used to fit curve to data points

β, βijk, βijk;q Coefficients used to fit αi;q to model parameters θ

f Function representing the simulator code (seismic, MT, orgravity)

h Function representing the parametric part of the emulator(seismic, MT, or gravity)

gq Theoretical Gaussian residual function for case q

Gq Computed approximation to gqLi Laguerre polynomial of order i

κq;n Weighting for data point n in computing plausibility condition

Superscripts p and qand w

Numbers of α and β coefficients and layers

Subscripts q ¼ x, ωr,ωι, ρ, ψ

Denote seismic, MT (real or imaginary), gravity, or seismicspike domains

Subscripts em, sim,obs, targ

Denote emulated, simulated, observed, and target values

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addition, in using an emulator-based method such as this, the sci-entist is ultimately trading simulator accuracy with the ability toscreen a large area of model space. Ultimately, for the problemand purpose at hand, it comes down to the questions of (1) howone can most efficiently extract sufficient information from the sys-tem as to which areas of the model space are plausible and which areimplausible, and (2) at what stage in the model-screening process itis optimal to use the information contained in the system. A keyavenue for investigation in this regard may lie in considering theway in which 3D earth models are parameterized. In this study, forexample, we reduced the number of traveltime data points by repre-senting the traveltime functions as polynomials parameterized by aseries of α coefficients, on the basis that in general, due to a highsampling density, there is a strong correlation between adjacenttraveltime measurements, and particularly in the early stages ofmodel-space screening, the resolution of or information content in thetraveltime data is much larger than is necessary to exclude a verylarge amount of model space. In a similar manner, at the early stagesof a typical model-screening process, the information content in ahigh-resolution 3D model is undoubtedly much larger than is neces-sary to exclude a very large region of model space. The natural way totake account of this may be to consider adaptive parameterization ofthe model space, for example, as considered by Trinks et al. (2005)and others, beginning with a coarse parameterization and includingmore information from the data set and introducing more modelparameters as the plausible model space is refined. Along with con-sidering parallelization of the emulator-building and screening proc-esses, we consider this to be fundamental to the future developmentand practical application of this method.

Emulator automation and tuning considerations

Although the construction and use of emulators give considerablecomputer runtime savings, in this study, the emulator constructionprocess has itself required considerable investment of user time andthought. In many ways, the designing of an emulator is never quite aone-size-fits-all scenario, for example, the class of functions chosento represent the data functions (in this case, Laguerre polynomials),the number of emulation cycles to cut off the analysis, the criterionfor a model to be considered plausible, or implausible, the numberof simulator runs to train an emulator within any given setting, andmany others. For a real-world scenario, formal sampling of thesetuning parameters would need to be carried out, including the in-corporation of insight from an expert geophysicist.This study has sought to demonstrate the potential for emulator-

type technology as an effective tool to facilitate the rapid screeningof model space. Although the choice of emulator design can never

really be divorced from a consideration of the particular scenario athand, through further development, including a more robust treat-ment of the prior model space, and the various tuning parameters,we envisage that it may be possible to develop a semiautomatedsystem for particular types of geophysical settings, parameterizableby, for example, a variety of function classes with which to fit thedata to the model parameters, coupling relationships, and implau-sibility criteria. However, at the moment, this kind of screening ap-proach still seems to be in its infancy in the world of geophysics, andso considerable further work is needed before this could be realizedand the method becomes commercially feasible.

Why use this kind of approach

In cases in which system uncertainty is large, and when there areseveral kinds of joint data sets, as in this example, the ability to dis-cern the full plausible model space greatly adds to the understandingof the system concerned. The advantage of a forward Monte Carloapproach such as that taken here is first that the whole prior plausiblemodel space is considered and second that joint models generatedusing different physical relationships can be tested. Based on the frac-tion of models generated using each relationship that are acceptedand rejected, conclusions can be made regarding the probability thata particular set of rock properties are present across the profile. This isdemonstrated by the salt probability map shown in Figure 20. It isnoticeable that although each of the velocity, resistivity, and densitymodels shown in Figures 17–19 may be liable to be individually mis-interpreted (and indeed here, each of these is an average model, andso they are not in themselves best fits to the data sets) the probabilitymap shows us that although a wide range of velocity, resistivity, anddensity models may individually fit the data sets, if one considers thequestion of which set of physical relationships are preferred acrossthe profile, the result in Figure 20 shows where the presence of salt ismost probable.We consider the ability to not just ask the question of what the

optimum velocity, density, or resistivity models are, and rely onintuition to then make judgments, but to ask the direct question“What is the probability, given the data and prior understanding,that salt exists across the profile?” to be extremely powerful. Thisstudy demonstrates that it represents a more robust, satisfactory, anduseful way of informing decisions than considering a central aver-age parameter model, about which there is some specified uncer-tainty, which is the currently accepted pseudo-standard in manysettings.Presenting the two results together, the probabilistic analysis de-

scribed in this paper, and the result from the deterministic inversion(Figure 21), provides a powerful tool for the geologic interpreter.

From the deterministic result, we have an opti-mum map for the rock properties, and fromprobabilistic modeling, we can make informedjudgments about the kind of rock that is presentand the uncertainty associated with drawing par-ticular inferences about the rock types present.The implementation of the probabilistic approachpresented here has considerable potential for de-velopment, technically in terms of incorporating2D and 3D information, visualisation in termsof viewing joint multimodal information, and interms of the design of the emulators used to inter-polate the model space.

Table 4. The n weights for data points used in computation of plausibilityconditions for seismic and MT screening.

Seismic traveltime number (n) κx;nMT pointnumber (n) κωfrιg;n

−10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 34, 38, 42,46, 50, 54, 58, 62, 66, 70, 76, 82, 88

1 all (1–20) 1

All others in the range 1–99 not listed above 0

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CONCLUSIONS

In this study, we have demonstrated the use of emulator technol-ogy in the field of geophysical constraint. Through an initial invest-ment of training simulator runs, we have built emulators for seismic,gravity, and MT systems, which give a rapid uncertainty-calibratedestimate of the forward code outputs. These emulators give a speedincrease of several orders of magnitude over the full forward codeand, as a result of the uncertainty calibration, can be used to screenlarge areas of model space for plausibility.The increased constraint using several methods in a joint regime

over using a single method has been seen in that each method (seis-mic, MT, and gravity) provides complementary information for theexclusion of an implausible model space.We have seen that in cases in which a multimodal distribution of

plausible model parameters is observed, it is inappropriate, mislead-ing, and incorrect to present the results as some central averagevalue about which there is an uncertainty, as is often consideredthe normal accepted practice in the field of geophysical imaging.Screening the whole of the model space, rather than a small part,

in a forward scheme such as this allows not simply for the deter-mination of an optimum model. Because the method enables thetrialing of alternative candidate rock-physics relationships (which,in many situations, characterize the rock itself, rather than one prop-erty of that rock), the question of “With what probability can we saythat a given kind of structure exists?” can be directly answered, asshown in Figure 20. Presented perhaps in conjunction with the re-sult of a deterministic parametric joint inversion, this represents avery powerful tool for the purpose of informing geoeconomic de-cisions, particularly in relation to risk.

ACKNOWLEDGMENTS

The authors would like to thank ITF and the sponsors of the JIBAConsortium, through which this work was funded, for their financialprovision and advice. JIBA sponsors are as follows: Statoil, Chev-ron, ExxonMobil, Nexen, Wintershall, RWE, and Shell. We wouldalso like to thank Statoil in particular for providing a data set onwhich to test the method. Most of the coding for this study wascarried out using R, an open-source statistical coding environment,along with several additional modules (R Development Core Team,2008; Carnell, 2009; Dutang, 2009). We are also grateful to twoanonymous reviewers for their comments on a previous version ofthis paper, and their suggestion to use Laguerre polynomials as ba-sis functions for fitting the data sets. Inspiration for the acceptanceratio maps of Figures 13–16 came from S. Pearse’s Ph.D. thesis.

APPENDIX A

SYMBOLOGY

The mathematical symbols used throughout this paper are de-fined in Table 3.

Seismic emulator

Here, t refers to seismic traveltime (first-arrival wide-angle re-fraction), x refers to the source-receiver offset, and vm and sm referto the velocity and thickness of layer m, respectively. The px refersto the order of polynomial used in fitting curves to each trainingdata set using the coefficients αx. Equation A-1 shows how we

rerepresent the traveltime curves as a set of polynomial coefficientsαi;x and a misfit function gxðxÞ. Equations A-2 and A-3 show howthe data coefficients αi;x are then represented as a polynomial in themodel parameters θx, parameterized by a further set of polynomialcoefficients βx, and a further misfit function gi;xðθxÞ:

t ¼�Xpx

i¼0

αi;xxie−xLiðxÞ�þ gxðxÞ; (A-1)

θx ¼ ½v1 v2 v3 v4 v5 v6 v7 s1 s2 s3 s4 s5 s6 s7 �T;(A-2)

and

αi;x ¼�Xwx

k¼1

Xqxj¼0

βijkθjk;xe

−θk;xLiðθk;xÞ�þ gi;xðθxÞ: (A-3)

Equations A-4–A-8 show the reorganization of the formulationdescribed in equations A-1–A-3 to compute the global emulatormisfit function Gðx; θxÞ. In practice, we calculate a variant of thisfunction GxðxÞ averaged over all model parameters. The qx and wx

are the order of polynomial used to write the data coefficients α as afunction of the model parameters θ and the number of model param-eters θ, respectively.

t ¼Xpx

i¼0

Xwx

k¼1

Xqxj¼0

βijkθjk;xe

−θk;xLiðθk;xÞxie−xLiðxÞ

þ gi;xðθxÞxie−xLiðxÞ þ gxðxÞ; (A-4)

¼Xpx

i¼0

Xwx

k¼1

Xqxj¼0

βijkθjk;xe

−θk;xLiðθk;xÞxie−xLiðxÞ

þ�Xpx

i¼0

ðgx;iðθxÞxie−xLiðxÞÞ þ gxðxÞ�; (A-5)

¼Xpx

i¼0

Xwx

k¼1

Xqxj¼0

βijkθjk;xe

−θk;xLiðθk;xÞxie−xLiðxÞ þ Gðx; θxÞ;

(A-6)

≈Xpx

i¼0

Xwx

k¼1

Xqxj¼0

βijkθjk;xe

−θk;xLiðθk;xÞxie−xLiðxÞ þ GxðxÞ;

(A-7)

and

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GxðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnmax

n¼1ðtem;nðxnÞ − tsim;nðxnÞÞ2nmax

s; (A-8)

where ðtn; xnÞ denote the individual (traveltime, offset) observationpoints.Equations A-13–A-30 show the analogs formulation for the MT

and gravity emulators. In each case, as with the seismic case, theemulators consist of a set of parametric coefficients βijk;ω or βjk;ρand an uncertainty function GωðωÞ or Gρ.

Spike emulator

ψ ¼�d2tdx2

�2

; (A-9)

xðψmaxiÞ ¼

�X3k¼1

Xqψj¼0

βijk;ψθjk;x

�þ gi;ψðθxÞ; (A-10)

≈�Xwψ

k¼1

Xqψj¼0

βijk;ψθjk;x

�þGψ ;i; (A-11)

and

Gψ ;i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnmax

n¼1 ðxðψ em;maxiÞ − xðψ sim;maxi

ÞÞ2nmax

s: (A-12)

MT emulator

R ¼ logðZrÞ ¼�Xpω

i¼0

αi;ωrðlogðωÞÞielogðωÞLiðlogðωÞÞ

�

þ grðωÞ; (A-13)

I ¼ logðZιÞ ¼�Xpω

i¼0

αi;ωιðlogðωÞÞielogðωÞLiðlogðωÞÞ

�

þ gιðωÞ; (A-14)

θMT ¼ ½r1 r2 r3 r4 r5 r6 r7 s1 s2 s3 s4 s5 s6 s7 �T;(A-15)

αi;ω ¼�Xwω

k¼1

Xqωj¼0

βijk;ωθjk;ωe

−θk;ωLjðθk;ωÞ�þ gi;ωðθωÞ;

(A-16)

R¼Xpωr

i¼0

Xwωr

k¼1

Xqωrj¼0

βijk;ωrθjk;ωe

θk;ωLjðθk;ωÞðlogðωÞÞielogðωÞLiðlogðωÞÞ

þXpωr

i¼0

ðgi;ωrðθωÞðlogðωÞÞielogðωÞLiðlogðωÞÞÞþgωrðωÞ; (A-17)

¼Xpωr

i¼0

Xwωr

k¼1

Xqωrj¼0

βijk;ωrθjk;ωe

θk;ωLjðθk;ωÞðlogðωÞÞielogðωÞLiðlogðωÞÞ

þ�Xpωr

i¼0

ðgi;ωrðθωÞðlogðωÞÞielogðωÞLiðlogðωÞÞÞþgωrðωÞ�;

(A-18)

¼Xpωr

i¼0

Xwωr

k¼1

Xqωrj¼0

βijk;ωrθjk;ωe

θk;ωLjðθk;ωÞ

× ðlogðωÞÞielogðωÞLiðlogðωÞÞ þGðω; θωÞ; (A-19)

≈Xpωr

i¼0

Xwωr

k¼1

Xqωrj¼0

βijk;ωrθjk;ωe

θk;ωLjðθk;ωÞ

× ðlogðωÞÞielogðωÞLiðlogðωÞÞ þ GωðωÞ; (A-20)

and

GωrðωÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnmax

n¼1ðRem;nðωnÞ − Rsim;nðωnÞÞ2nmax

s; (A-21)

where ðωn; RnÞ denotes individual (frequency, impedance) observa-tion points.

I¼Xpωι

i¼0

Xwωι

k¼1

Xqωιj¼0

βijk;ωθjk;ωe

θk;ωLjðθk;ωÞðlogðωÞÞielogðωÞLiðlogðωÞÞ

þXpωι

i¼0

ðþgi;ωιðθωÞðlogðωÞÞielogðωÞLiðlogðωÞÞÞþgωðωÞ;

(A-22)

¼Xpωι

i¼0

Xwωι

k¼1

Xqωιj¼0

βijk;ωιθjk;ωe

θk;ωLjðθk;ωÞðlogðωÞÞielogðωÞLiðlogðωÞÞ

þ�Xpωι

i¼0

ðgi;ωιðθωιÞðlogðωÞÞielogðωÞLiðlogðωÞÞÞþgωðωÞ�;

(A-23)

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¼Xpωι

i¼0

Xwωι

k¼1

Xqωj¼0

βijk;ωιθjk;ωe

θk;ωLjðθk;ωÞ

× ðlogðωÞÞielogðωÞLiðlogðωÞÞ þ Gðω; θωÞ; (A-24)

≈Xpωι

i¼0

Xwωι

k¼1

Xqωιj¼0

βijk;ωιθjk;ωe

θk;ωLjðθk;ωÞ

× ðlogðωÞÞielogðωÞLiðlogðωÞÞ þGωðωÞ; (A-25)

and

GωιðωÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnmax

n¼1 ðIem;nðωnÞ − Isim;nðωnÞÞ2nmax

s; (A-26)

where ðωn; InÞ denotes individual (frequency, impedance) observa-tion points.

Gravity emulator

θρ ¼ ½ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 s1 s2 s3 s4 s5 s6 s7 �T;(A-27)

ϕ ¼�Xwρ

k¼1

Xqρj¼0

βjk;ρθjk;ρ

�þ gρðθρÞ; (A-28)

≈Xwρ

k¼1

Xqρj¼0

βjk;ρθjk;ρ þ Gρ; (A-29)

and

Gρ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnmax

n¼1 ðϕem;n − ϕsim;nÞ2nmax

s; (A-30)

where ϕn denotes individual gravity observations.

Plausibility conditions

The seismic screening plausibility condition is as follows:

Xnmax

n¼1

κx;nmax½jðtemðxnÞ − tobsðxnÞÞj − γxGxðxnÞ; 0�

GxðxnÞPnmax

p¼1 κx;p< nmax:

(A-31)

The weights κx;n are shown in Table 4.For the seismic spike screening, the plausibility condition that

must be met for each of the i spikes is as follows:

jxemðψmaxiÞ − xobsðψmaxi

Þj − γψGψ ;i < 0: (A-32)

For the MT screening, the plausibility condition is that both of thefollowing conditions must be met:

Xnmax

n¼1

κωr;nmax½jðRemðωnÞ−RobsðωnÞÞj−γωGωrðωnÞ;0�

GωrðωnÞPnmax

p¼1κωr;p<nmax;

(A-33)

and

Xnmax

n¼1

κωι;nmax½jðIemðωnÞ− IobsðωnÞÞj−γωGωιðωnÞ;0�

GωιðωnÞPnmax

p¼1 κωι;p<nmax;

(A-34)

and The weights κωfrιg;n are shown in Table 4.The gravity screening plausibility condition is as follows:

jϕem − ϕobsj − γρGρ < 0: (A-35)

Joint sampling method

The methodology for generating a new set of velocity and thick-ness values θ 0

i from the joint parameter distribution from the pre-vious screening cycle (where the parameter values are given by θi)is shown in equation A-36. The values θi;max and θi;min are the maxi-mum and minimum bounds of θ from the previous cycle as follows:

θ 0i ¼ θi þ Uð−0.01ðθi;max − θi;minÞ; 0.01ðθi;max − θi;minÞÞ:

(A-36)

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